Tag Probability

Brain teasers and puzzles involving randomness, probability, and statistics

The Monty Hall Problem Explained

The Monty Hall Problem

You are on a game show in which there are three identical doors, one with a car behind it and two with goats behind them. You must pick one door, and you win if that door has the car behind it.

After you pick a door, the host of the game show always opens a door you didn’t choose that has a goat behind it. This leaves the door you chose and one other remaining door, and you are given the option to switch your choice to the other remaining door.

Should you switch or should you stick to your original choice? What chance of winning would that give you?

The History

The Monty Hall Problem is a classic probability puzzle, named for its similarity to the game show “Let’s Make a Deal”, which was hosted by Monty Hall. The problem was made famous when Marilyn vos Savant answered it correctly in her column in a popular magazine, and thousands of readers wrote letters to the magazine arguing her solution was wrong!

The solution can be counter-intuitive, so give it some thought and then scroll down to see the Monty Hall Problem explained.

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Russian Roulette Riddle

In the morbid game of Russian Roulette, a partially loaded revolver with a six-chamber cylinder is randomly spun, pointed at one of the players, and fired. If the revolver landed on an empty chamber, the lucky player is safe, and the process is repeated with the next player. The obvious objective of the game is to not get shot.

You find yourself stuck in a game of Russian Roulette. A freshly loaded revolver is aimed at the first player, and it turns out to be an empty chamber. Your turn is next, and you are given the choice to either:

  • Spin the cylinder before pulling the trigger (i.e., you get a random new chamber)
  • Or just pull the trigger (i.e., let the revolver fire whatever is in the next chamber)

Which choice should you pick if the revolver was originally:

  1. Loaded with one bullet?
  2. Loaded with bullets in two random chambers?
  3. Loaded with bullets in two consecutive chambers?

Assume the revolver cannot misfire, and that spinning the cylinder lands on all chambers with equal probability.

Some variation of this Russian Roulette riddle was once asked in interviews at Jane Street, Susquehanna International Group (SIG), Facebook (now Meta), UBS, Capital One, and more.

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A truel is a three-way duel. The three participants take turns firing one shot at whichever opponent they choose, until only one is remaining.

Allison, Ben, and Chase are in a truel, and have varying degrees of accuracy: Allison has a 50% chance of hitting her intended target, Ben has a 80% chance, and Chase has a 100% chance. Allison gets to shoot first, then Ben, then Chase, and repeating in that order until only one person is remaining.

Assume that the accuracy of all participants are publicly known, and everyone is trying to maximize their chances of winning. What is Allison’s optimal strategy, and what is her likelihood of winning under that strategy?

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Reroll the Die

Suppose there is a game in which you roll a fair, 6-sided die and win dollars equal to the outcome of the roll. How much would you expect to win on average?

Suppose, if you don’t like the outcome of the roll, you can reroll the die once, and win dollars equal to the outcome of the 2nd roll (once you choose to reroll, you can no longer go back to the 1st roll). How much would you expect to win on average?

Suppose, if you don’t like the outcome of the 2nd roll, you can reroll the die once more, and win dollars equal to the outcome of the 3rd roll (once you choose to reroll, you can no longer go back to previous rolls). How much would you expect to win on average?

This was an actual brain teaser question once asked at Jane Street for an interview for an intern role.

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Vacant Room Probability

Your workplace has a phone room for employees to quietly make personal calls. The room has no windows, just a sign that can be switched from “Vacant” to “Occupied”. However, employees differ in how consistently they use the sign:

  • 1/2 of them always switch to “Occupied” when they enter and “Vacant” when they exit.
  • 1/4 of them ignore the sign altogether – the sign will always read the same before, during, and after their visit.
  • 1/4 of them always switching to “Occupied” when they enter, but always forget to switch back to “Vacant” when they exit.

If the room is actually occupied exactly 1/2 of the time, what is the probability the room is actually vacant when the sign reads “Vacant”?

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Tale of Two Trains

There are two trains that run between two cities. The trains are identical and run on identical routes, so passengers have no preference between the two and would take whichever train that pulls into the station. The trains run at the same frequency: exactly once an hour.

You often travel between the two cities on a whim, and when you do so, you show up at the station at a completely random time. Yet after many trips over the years, you notice that you have taken one of the trains three times as often as the other. Is this just really bad/good luck, or is there another likely explanation?

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